62 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS. 



6. Helices and Spirals of Archimedes. 



Mature cylindrical axes exhibiting spiral phyllotaxis in which 

 the lateral members are closely set and of uniform character, 

 present remarkably beautiful appearances of helices with parallel 

 screw-threads winding in converse directions (cf. figs. StaTigeria, 

 (29), Cereus (30), Uuphorbia (31), Araucaria (32).). 



Such spirals with equidistant coils continued upwards on a cone, 

 would on the unrolled surface constitute portions of Archimedean 

 spirals as pointed out by the Bravais, and the projection on a trans- 

 verse plane would similarly give intersecting Archimedean spirals. 



The fact that similar helices are produced by torsion action 

 apparently forms the basis of all torsion theories of phyllotaxis, 

 whether in the obvious form of Airy's hypothesis or in the veiled dis- 

 placement system of Schwendener. As in other instances, however, 

 the same effect may be produced by widely different causes, but the 

 fact that the curves exhibited in phyllotaxis in the horizontal 

 plan may be spirals of Archimedes leading on to helices on the 

 cylindrical stem has been very generally accepted, and repre- 

 sented in diagrams in which concentric circles are taken in 

 arithmetical progression. 



So far, in fact, as such curves can be judged by the eye the 

 approximation is very close, and not only so, but the curve drawn 

 on a specimen (cf. figs. 2, 3, 4) is clearly more like such a construc- 

 tion than the theoretical log. spiral system previously postulated. 



Further consideration, however, shows wide differences ; thus it is 

 clear to begin with, that the phyllotaxis helices observed on a shoot 

 are not torsion spirals in any sense, but are merely the result of a 

 uniform development in both lateral member and internode 

 whereby a certain constant volume is reached and then further 

 growth is checked. The helices are thus not produced by the uni- 

 form growth of all the lateral members which are initiated at 

 different times, and would, if the rate of growth were constant in 

 all, remain always unequal, but they are the result of a progressive 

 cessation of growth, — that is to say, the helices are of secondary 

 origin, and any spiral series of members, whatever the primary 



